Incompressible surfaces and pseudo-Anosov flows
type: Item
instance of: scholarly article
title: Incompressible surfaces and pseudo-Anosov flows
zbMATH Open document ID: 0927.57009
DOI: 10.1016/S0166-8641(97)00120-X
author: Brian S. Mangum
published in: Topology and its Applications
publication date: 1999-12-08T00:00:00Z
review text: A dichotomoy exists between \(\pi_1\)-injective maps \(g\) of a closed surface \(S\) into a hyperbolic 3-manifold \(M\): by results of Bonahon, Marden and Thurston, either there is a finite covering space \(\widehat M\rightarrow M\) and a lift \(\widehat g:S\rightarrow\widehat M\) which is the inclusion of a fiber of a fibration \(\widehat M\rightarrow S^1\) or \(g_*\pi_1(S)\subset\pi_1(M)\) is quasi-Fuchsian. In the first case \(g_*\pi_1(S)\) is a geometrically infinitely discrete group of isometries of hyperbolic space and the second case \(g_*\pi_1(S)\) is geometrically finite. The present paper gives criteria for the geometrically infinite case in terms of a quasi-geodesic pseudo-Anosov flow \(\Phi\) on \(M\) which satisfies two additional global conditions. The map \(g\) is assumed to be an immersion transverse to \(\Phi\). Then \(g_*\pi_1(S)\) is geometrically infinite if and only if every lift \(\widetilde g:\widetilde S\rightarrow\widetilde M\) meets every orbit of the lifted flow on \(\widetilde M\). In that case \(g\) lifts to an embedded fiber and \(\Phi\) lifts to a suspension flow. The paper is clearly and carefully written. The methods of proof involve work of Fried and Mosher on cross-sections to pseudo-Anosov flows, and work of Gabai and Thurston on the norm of homology of 3-manifolds. Several interesting dynamical properties of these flows are discussed as well.
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 37C85
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 57N35
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 1307965
zbMATH Keywords: hyperbolic 3-manifold
zbMATH Keywords: pseudo-Anosov diffeomorphism
zbMATH Keywords: suspension flow
zbMATH Keywords: mapping torus
zbMATH Keywords: quasi-Fuchsian group
zbMATH Keywords: Thurston norm
zbMATH Keywords: quasigeodesic
zbMATH Keywords: geometric finiteness
zbMATH Keywords: bundles over \(S^1\)
zbMATH Keywords: incompressible surface
reviewed by: William M. Goldman
MaRDI profile type: Publication
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full work available at URL: s0166-8641(97)00120-x
OpenAlex ID: W2082134285
description: scientific article; zbMATH DE number 1307965